In this paper we obtain some operator inequalities for functions

defined by power series with real coefficients and, more specifically, with non-

negative coefficients. In order to obtain these inequalities some recent reverses

of the Schwarz inequality for vectors in inner product spaces are utilized. Natural applications for some elementary functions of interest are also provided

Dragomir, Sever S

English

In this paper, we obtain some operator inequalities for functions defined by power series with real coefficients and, more specifically, with nonnegative coefficients. In order to obtain these inequalities, some recent reverses of the Schwarz inequality for vectors in inner product spaces are utilized. Natural applications for some elementary functions of interest are also provided

Victoria University Institutional Repository (VUIR)

Some inequalities for power series of selfadjoint operators in Hilbert spaces via reverses of the Schwarz inequality

Victoria University Eprints Repository

SOME INEQUALITIES FOR POWER SERIES OF SELFADJOINTOPERATORS IN HILBERT SPACES VIA REVERSES OF THESCHWARZ INEQUALITYS.S. DRAGOMIRAbstract. In this paper we obtain some operator inequalities for functionsdened by power series with real coe¢ cients and, more specically, with non-negative coe¢ cients. In order to obtain these inequalities some recent reversesof the Schwarz inequality for vectors in inner product spaces are utilized. Nat-ural applications for some elementary functions of interest are also provided.1. IntroductionLet f (z) :=P1n=0 anzn be an analytic function dened by a power series withnonnegative coe¢ cients an; n  0 and convergent on the open disk D (0; R)  C,R > 0: As the most natural examples of such functions we havef (z) =11  z =1Xn=0zn; z 2 D (0; 1) and f (z) = exp (z) =1Xn=01n!zn; z 2 C:Other function as power series representations with nonnegative coe¢ cients are,for instancecosh (z) =1Xn=01(2n)!z2n; z 2 C;sinh (z) =1Xn=01(2n+ 1)!z2n+1; z 2 C;12ln1 + z1  z=1Xn=112n  1z2n 1; z 2 D (0; 1) ;sin 1 (z) =1Xn=0  n+ 12p (2n+ 1)n!z2n+1; z 2 D (0; 1) ;tanh 1 (z) =1Xn=112n  1z2n 1; z 2 D (0; 1) ;2F1 (; ; ; z) =1Xn=0  (n+ )   (n+ )   ()n!  ()   ()   (n+ )zn; ; ;  > 0; z 2 D (0; 1) :Date : 19 January, 2008.1991 Mathematics Subject Classication. Primary 47A12, 47A30; Secondary 47A63.Key words and phrases. Bounded linear operators, Commutators, Cartesian decomposition,Accretive operators.12 S.S. DRAGOMIRNow, by the help of the power series f (z) =P1n=0 anzn we can naturally con-struct another power series which will have as coe¢ cients the absolute values ofthe coe¢ cient of the original series, namely, fA (z) :=P1n=0 janj zn: It is obviousthat this new power series will have the same radius of convergence as the originalseries.As some natural examples that are useful for applications, we can point out that,iff (z) =1Xn=1( 1)nnzn = ln11 + z; z 2 D (0; 1) ;g (z) =1Xn=0( 1)n(2n)!z2n = cos z; z 2 C;h (z) =1Xn=0( 1)n(2n+ 1)!z2n+1 = sin z; z 2 C;l (z) =1Xn=0( 1)n zn = 11 + z; z 2 D (0; 1) ;then the corresponding functions constructed by the use of the absolute values ofthe coe¢ cients are obviouslyfA (z) = ln11  z ; gA (z) = cosh z;hA (z) = sinh z and lA (z) =11  zand they are dened on the same domain as the generating functions.Before we are able to state the new inequalities for functions of selfadjoint oper-ators we need to recall the continuous functional calculus for such operators.Let A be a selfadjoint linear operator on a complex Hilbert space (H; h:; :i) :The Gelfand map establishes a -isometrically isomorphism  between the setC (Sp (A)) of all continuous functions dened on the spectrum of A; denoted Sp (A) ;an the C-algebra C (A) generated by A and the identity operator 1H on H asfollows (see for instance [7, p. 3]):For any f; g 2 C (Sp (A)) and any ;  2 C we have(i)  (f + g) =  (f) +  (g) ;(ii)  (fg) =  (f)  (g) and  f= (f);(iii) k (f)k = kfk := supt2Sp(A) jf (t)j ;(iv)  (f0) = 1H and  (f1) = A; where f0 (t) = 1 and f1 (t) = t; for t 2 Sp (A) :With this notation we denef (A) :=  (f) for all f 2 C (Sp (A))and we call it the continuous functional calculus for a selfadjoint operator A:If A is a selfadjoint operator and f is a real valued continuous function on Sp (A),then f (t)  0 for any t 2 Sp (A) implies that f (A)  0; i:e: f (A) is a positiveoperator on H: Moreover, if both f and g are real valued functions on Sp (A) thenthe following important property holds:(P) f (t)  g (t) for any t 2 Sp (A) implies that f (A)  g (A)in the operator order of B (H) :SOME INEQUALITIES FOR POWER SERIES 3For a recent monograph devoted to various inequalities for functions of selfadjointoperators, see [7] and the references therein. For other results, see [9], [10], [11] and[8].The main aim of the present paper is to provide some operator inequalities forfunctions dened by power series with real coe¢ cients and, more specically, withnonnegative coe¢ cients. In order to obtain these inequalities some recent reversesof the Schwarz inequality for vectors in inner product spaces are utilized. Naturalapplications for some elementary functions of interest are also considered.2. Some Operator InequalitiesWe use the following vector inequality obtained by the author in [1] (see also [5,pp. 28-29]):Lemma 1. Let (H; h; i) be an inner product space over K and z; y 2 H; ;  2 K(  6=  ) so that either(2.1) Re h y   z; z   yi  0;or, equivalently,(2.2)z    +  2 y  12 j   j kykholds. Then we have the inequalities(2.3) (0 ) kzk kyk   jhz; yij  kzk kyk  Re    + j  + j hz; yi kzk kyk   Re   + j  + j hz; yi 14 j   j2j  + j kyk2:The constant 14 in the last inequality is best possible.Perhaps a more useful particular case of the above lemma isCorollary 1. Let (H; h; i) be an inner product space over K and z; y 2 H; N >n > 0 so that either(2.4) Re hNy   z; z   nyi  0;or, equivalently,(2.5)z   n+N2 y  12 (N   n) kykholds. Then we have the inequalities(2.6) (0 ) kzk kyk   jhz; yij  kzk kyk   jRe hz; yij kzk kyk   Re hz; yi  14 (N   n)2n+Nkyk2 :The constant 14 in the last inequality is best possible.The following result holds:4 S.S. DRAGOMIRTheorem 1. Let consider the power series f (z) =P1n=0 anzn with real coe¢ cientsan that is convergent on the open disk D (0; R) with R > 0: If the selfadjoint operatorT on the Hilbert space H has the spectrum Sp (T )  [m;M ] and 0 < m < M withM2pmM< R, then for any x 2 H with kxk = 1 we have the inequality(2.7) max fkf (T )xk ; kfA (T )xkg  hfA (T )x; xi+18fAM2pmM  2fApmM+ fAm2pmM:Proof. Since Sp (T )  [m;M ] then we have 0 < mkI  T k MkI in the operatororder of B (H) and for any natural number k; which implies thatMkx  T kx; T kx mkx  0or, equivalently, T kx  mk +Mk2 x  12  Mk  mkfor any x 2 H with kxk = 1:Now, if we apply the inequality (2.6), then we get(2.8)T kx  T kx; x+ 14 (Mk  mk)2Mk +mkfor any natural number k and for any x 2 H with kxk = 1:Since Mk +mk  2pmMk; then we have(2.9)(Mk  mk)2Mk +mk M2k   2mkMk +m2k2pmMk=12"M2pmMk  2pmMk+m2pmMk#for any natural number k:Utilising (2.8) and (2.9) we get(2.10)T kx  T kx; x+ 18"M2pmMk  2pmMk+m2pmMk#for any natural number k and for any x 2 H with kxk = 1:Now, if we multiply (2.10) with jakj sum over k from 0 to n and use the triangleinequality, we deduce(2.11) max(nXk=0akTkx ;nXk=0jakjT kx)nXk=0jakjT kx  * nXk=0jakjT kx; x++18"nXk=0jakjM2pmMk  2nXk=0jakjpmMk+nXk=0jakjm2pmMk#for any natural number n and for any x 2 H with kxk = 1:SOME INEQUALITIES FOR POWER SERIES 5Now, observe that0 <m2pmM<pmM < M <M2pmM< R;therefore all the series in the equation (2.11) are convergent and taking the limitover n!1 in (2.11) we deduce the desired result (2.7). The following inequality also holds [2] (see also [5, p. 21]):Lemma 2. Let (H; h; i) be an inner product space over K and z; y 2 H; ;  2 Kwith Re ( ) > 0 so that either (2.1) or, equivalently, (2.2) holds true. Then wehave the inequalities(2.12) kxk kyk  12 Re   +  hx; yiRe ( ) 12 j  + jRe ( )jhx; yij :The constant 12 is best possible in both inequalities.Corollary 2. Let (H; h; i) be an inner product space over K and z; y 2 H; N >n > 0 so that either (2.4) or, equivalently, (2.5) holds true. Then we have(2.13) kxk kyk  12 N + npNnRe hx; yi :The constant 12 is best possible.We are able to provide now another upper bound for kf (T )xk with x 2 H,kxk = 1 as follows:Theorem 2. Let consider the power series f (z) =P1n=0 anzn with real coe¢ cientsan that is convergent on the open disk D (0; R) with R > 0: If the selfadjoint operatorT on the Hilbert space H has the spectrum Sp (T )  [m;M ] and 0 < m < M withMqMm < R, then for any x 2 H with kxk = 1 we have the inequality(2.14) max fkf (T )xk ; kfA (T )xkg 12*"fA rMm T!+ fArmM T#x; x+:Proof. Since Sp (T )  [m;M ] thenMkx  T kx; T kx mkx  0for any x 2 H with kxk = 1:Now, if we apply the inequality (2.13), then we can state that(2.15)T kx  12 Mk +mkpMkmkT kx; x=1224 rMm!k+rmMk35T kx; xfor any natural number k and for any x 2 H with kxk = 1:Now, if we multiply (2.10) with jakj sum over k from 0 to n and use the triangleinequality, we deduce6 S.S. DRAGOMIR(2.16) max(nXk=0akTkx ;nXk=0jakjT kx)nXk=0jakjT kx  * nXk=0jakjT kx; x+ 12*nXk=0jakj rMm!kT kx+nXk=0jakjrmMkT kx; x+for any natural number n and for any x 2 H with kxk = 1:SinceMrmM< M < MrMm< Rtherefore all the series in the equation (2.16) are convergent and taking the limitover n!1 in (2.16) we deduce the desired result (2.14). In order to obtain another reverse inequality for the operator f (T ) we need thefollowing result [3] (see also [5, p. 4]):Lemma 3. Let (H; h; i) be an inner product space over K and z; y 2 H; ;  2 Kso that either (2.1) or, equivalently, (2.2) holds true. Then we have the inequality(2.17) (0 ) kxk2 kyk2   jhx; yij2  14 j   j2 kyk4 :The constant 14 is best possible.The case for positive bounds that will be used in the following is incorporated inCorollary 3. Let (H; h; i) be an inner product space over K and z; y 2 H; N >n > 0 so that either (2.4) or, equivalently, (2.5) holds true. Then we have(2.18) (0 ) kxk2 kyk2   jhx; yij2  14 (N   n)2 kyk4 :The constant 14 is best possible.The following result also holds:Theorem 3. Let consider the power series f (z) =P1n=0 anzn with real coe¢ cientsan that is convergent on the open disk D (0; R) with R > 0: If the selfadjoint operatorT on the Hilbert space H has the spectrum Sp (T )  [m;M ] and 0 < m < M withM < R, then for any x 2 H with kxk = 1 we have the inequality(2.19) max fkf (T )xk ; kfA (T )xkg  hfA (T )x; xi+ 12[fA (M)  fA (m)] :Proof. Since Sp (T )  [m;M ] thenMkx  T kx; T kx mkx  0for any x 2 H with kxk = 1:Observe that from (2.18) we getkxk kyk rjhx; yij2 + 14 (N   n)2 kyk4  jhx; yij+ 12 (N   n) kyk2SOME INEQUALITIES FOR POWER SERIES 7provided that either (2.4) or, equivalently, (2.5) holds true.Now if we apply this inequality for T k we can write thatT kx  T kx; x+ 12  Mk  mkfor any natural number k and for any x 2 H with kxk = 1:Finally, if we apply a similar argument to the one from the above theorems wederive the desired result (2.19). The details are omitted. Finally, in order to obtain the last reverse inequality, we need to use the followingKlamkin-McLenaghan inequality for vectors in inner product spaces that has beenobtained in [6]:Lemma 4. Let (H; h; i) be an inner product space over K and z; y 2 H; ;  2 Kwith Re ( ) > 0 so that either (2.1) or, equivalently, (2.2) holds true. Then wehave the inequality(2.20) (0 ) kxk2 kyk2   jhx; yij2 hj  + j   2pRe ( )ijhx; yij kyk2 :As a particular case of interest we haveCorollary 4. Let (H; h; i) be an inner product space over K and z; y 2 H; N >n > 0 so that either (2.4) or, equivalently, (2.5) holds true. Then we have(2.21) (0 ) kxk2 kyk2   jhx; yij2 pN  pn2jhx; yij kyk2 :For a generalisation of (2.12), other similar results and the equlity case analysis,see [4].Theorem 4. Let consider the power series f (z) =P1n=0 anzn with real coe¢ cientsan that is convergent on the open disk D (0; R) with R > 0: If the selfadjoint operatorT on the Hilbert space H has the spectrum Sp (T )  [m;M ] and 0 < m < M withM < R, then for any x 2 H with kxk = 1 we have the inequality(2.22) max fkf (T )xk ; kfA (T )xkg  hfA (T )x; xi+ fA (M)  fApmM:Proof. Notice that from the inequality (2.21) we havekxk kyk rjhx; yij2 +pN  pn2jhx; yij kyk2 jhx; yij+pN  pnpjhx; yij kykprovided that either (2.4) or, equivalently, (2.5) holds true.Now if we apply this inequality for T k we can write thatT kx  T kx; x+ pMk  pmkqjhT kx; xij T kx; x+ pMk  pmkpMkfor any natural number k and for any x 2 H with kxk = 1:Finally, if we apply a similar argument to the one from the above theorems wederive the desired result (2.19). The details are omitted. 8 S.S. DRAGOMIRProblem 1. It is an open problem whether or not the above inequalities (2.7),(2.14), (2.19) and (2.22) are sharp, and if they are not, which bound is better fordi¤erent examples of functions.3. Applications1. Consider the function f (z) = 11 z =P1n=0 zn; z 2 D (0; 1) : If T is a boundedlinear operator on the Hilbert space (X; h; i) with the spectrum Sp (T )  [m;M ]and 0 < M3 < m < M < 1 then by the inequality (2.7) we have(3.1) (0 )(I   T ) 1 x  D(I   T ) 1 x; xE 18" pmMpmM  M2  21 pmM +pmMpmM  m2#while by (2.13) we have(3.2)(I   T ) 1 x  12*24 I  rMm T! 1x+I  rmM T 1x35 ; x+for any x 2 H with kxk = 1:If 0 < m < M < 1; then by (2.19) and (2.22) we have the simpler inequalities(3.3) (0 )(I   T ) 1 x  D(I   T ) 1 x; xE  12 M  m(1 m) (1 M)and(3.4) (0 )(I   T ) 1 x  D(I   T ) 1 x; xE  M  pmM(1 M)1 pmMrespectively, for any x 2 H with kxk = 1:2. Consider the function f (z) = ln11 z=P1n=1znn ; z 2 D (0; 1) : If Tis a bounded linear operator on the Hilbert space (X; h; i) with the spectrumSp (T )  [m;M ] and 0 < M3 < m < M < 1 then by the inequality (2.7) we have(3.5) (0 )ln (I   T ) 1 x  Dln (I   T ) 1 x; xE 18 ln264 mM1 pmM2pmM  M2pmM  m2375while by (2.13) we have(3.6)ln (I   T ) 1 x  12*ln24 I  rMm T! 1I  rmM T 135x; x+for any x 2 H with kxk = 1:If 0 < m < M < 1; then by (2.19) and (2.22) we have the simpler inequalities(3.7) (0 )ln (I   T ) 1 x  Dln (I   T ) 1 x; xE  12 ln1 m1 MSOME INEQUALITIES FOR POWER SERIES 9and(3.8) (0 )ln (I   T ) 1 x  Dln (I   T ) 1 x; xE  ln 1 pmM1 M!respectively, for any x 2 H with kxk = 1:3. Consider the exponential function f (z) = exp (z) =P1n=0znn! ; z 2 C. IfT is a bounded linear operator on the Hilbert space (X; h; i) with the spectrumSp (T )  [m;M ] and 0 < m < M <1 then by the inequality (2.7) we have(3.9) (0 ) kexp (T )xk   hexp (T )x; xi 18expM2pmM  2 exppmM+ expm2pmMwhile by (2.13) we have(3.10) kexp (T )xk  12*"exp rMm T!+ exprmM T#x; x+for any x 2 H with kxk = 1:Also, by (2.19) and (2.22) we have the simpler inequalities(3.11) (0 ) kexp (T )xk   hexp (T )x; xi  12 [exp (M)  exp (m)]and(3.12) (0 ) kexp (T )xk   hexp (T )x; xi  exp (M)  exppmMfor any x 2 H with kxk = 1:4. Consider the function f (z) = cos z =P1n=0( 1)n(2n)! z2n; z 2 C: Obviously, wehave fA (z) = cosh z =P1n=01(2n)!z2n; z 2 C: If T is a bounded linear operator onthe Hilbert space (X; h; i) with the spectrum Sp (T )  [m;M ] and 0 < m < M <1 then by the inequality (2.7) we have(3.13) kcos (T )xk  hcosh (T )x; xi+18coshM2pmM  2 coshpmM+ coshm2pmMwhile by (2.13) we have(3.14) kcos (T )xk  12*"cosh rMm T!+ coshrmM T#x; x+for any x 2 H with kxk = 1:Moreover, by (2.19) and (2.22) we have the simpler inequalities(3.15) kcos (T )xk  hcosh (T )x; xi+ 12 [cosh (M)  cosh (m)]and(3.16) kcos (T )xk  hcosh (T )x; xi+ cosh (M)  coshpmMrespectively, for any x 2 H with kxk = 1:10 S.S. DRAGOMIRIf we apply the same inequalities only for the function with positive coe¢ cientscosh then we also have(3.17) 0  kcosh (T )xk   hcosh (T )x; xi 18coshM2pmM  2 coshpmM+ coshm2pmMand(3.18) kcosh (T )xk  12*"cosh rMm T!+ coshrmM T#x; x+and(3.19) 0  kcosh (T )xk   hcosh (T )x; xi  12 [cosh (M)  cosh (m)]and(3.20) 0  kcosh (T )xk   hcosh (T )x; xi  cosh (M)  coshpmMrespectively, for any x 2 H with kxk = 1:Remark 1. The reader may obtain other operator inequalities by choosing appro-priate examples of functions dened by series with positive coe¢ cients. However,the details are omitted.References[1] S.S. Dragomir, New reverses of Schwarz, triangle and Bessel inequalities in inner prod-uct spaces, Austral. J. Math. Anal. Appl. 1(2004), No. 1, Art. 1, pp. 1-18. [Onlinehttp://ajmaa.org/searchroot/files/pdf/nrstbiips.pdf][2] S.S. Dragomir, Reverses of Schwarz, triangle and Bessel inequalities in inner prod-uct spaces, J. Inequal. Pure & Appl. Math. 5(2004), No. 3, Art. 76, pp. 1-18.[http://jipam.vu.edu.au/article.php?sid=432].[3] S.S. Dragomir, A counterpart of Schwarzs inequality in inner product spaces, East AsianMath. J., 20(1) (2004), 1-10.[4] S.S. Dragomir, Reverses of the Schwarz inequality generalising a Klamkin-McLenaghan result,Bull. Austral. Math. Soc., 73(1) (2006), 69-78.[5] S.S. Dragomir, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in InnerProduct Spaces. Nova Science Publishers, Inc., Hauppauge, NY, 2005. viii+249 pp. ISBN:1-59454-202-3[6] N. Elezovi´c, L. Maranguni´c and J. Peµcari´c, Unied treatment of complemented Schwarz andGrüss inequalities in inner product spaces, Math. Ineq. Appl. 8(2005), No. 2, 223-232.[7] T. Furuta, J. Mi´ci´c Hot, J. Peµcari´c and Y. Seo, Mond-Peµcari´c Method in Operator Inequal-ities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb,2005.[8] A. Matkovi´c, J. Peµcari´c and I. Peri´c, A variant of Jensens inequality of Mercers type foroperators with applications. Linear Algebra Appl. 418 (2006), no. 2-3, 551564.[9] B. Mond and J. Peµcari´c, Convex inequalities in Hilbert space, Houston J. Math., 19(1993),405-420.[10] B. Mond and J. Peµcari´c, On some operator inequalities, Indian J. Math., 35(1993), 221-232.[11] B. Mond and J. Peµcari´c, Classical inequalities for matrix functions, Utilitas Math., 46(1994),155-166.Research Group in Mathematical Inequalities & Applications, School of Engineering& Science, Victoria University, PO Box 14428, Melbourne VIC 8001, Australia.E-mail address : sever.dragomir@vu.edu.auURL: http://rgmia.vu.edu.au/dragomir/

Some Inequalities for Power Series of Selfadjoint Operators in Hilbert Spaces via Reverses of the Schwarz Inequality

In this paper we obtain some operator inequalities for functions\ud
defined by power series with real coefficients and, more specifically, with non-\ud
negative coefficients. In order to obtain these inequalities some recent reverses\ud
of the Schwarz inequality for vectors in inner product spaces are utilized. Natural applications for some elementary functions of interest are also provided

Some Inequalities for Power Series of Selfadjoint Operators in Hilbert Spaces via Reverses of the Schwarz Inequality

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